# Position-Dependent Mass Quantum systems and ADM formalism

### Submission summary

 As Contributors: Davood Momeni Preprint link: scipost_202008_00002v1 Date submitted: 2020-08-03 15:57 Submitted by: Momeni, Davood Submitted to: SciPost Physics Proceedings Proceedings issue: 4th International Conference on Holography, String Theory and Discrete Approach in Hanoi Academic field: Physics Specialties: Gravitation, Cosmology and Astroparticle Physics High-Energy Physics - Theory Quantum Physics Approach: Theoretical

### Abstract

The classical Einstein-Hilbert (EH) action for general relativity (GR) is shown to be formally analogous to the classical system with position-dependent mass (PDM) models. The analogy is developed and used to build the covariant classical Hamiltonian as well as defining an alternative phase portrait for GR. The set of associated Hamilton's equations in the phase space is presented as a first order system dual to the Einstein field equations. Following the principles of quantum mechanics,I build a canonical theory for the classical general. A fully consistent quantum Hamiltonian for GR is constructed based on adopting a high dimensional phase space. It is observed that the functional wave equation is timeless. As a direct application, I present an alternative wave equation for quantum cosmology. In comparison to the standard Arnowitt-Deser-Misner(ADM) decomposition and qunatum gravity proposals, I extended my analysis beyond the covariant regime when the metric is decomposed in to the $3+1$ dimensional ADM decomposition. I showed that an equal dimensional phase space can be obtained if one apply ADM decomposed metric.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

Resubmission 2008.02113v2 on 29 October 2020

Submission scipost_202008_00002v1 on 3 August 2020

## Reports on this Submission

### Anonymous Report 1 on 2020-10-21 (Invited Report)

• Cite as: Anonymous, Report on arXiv:scipost_202008_00002v1, delivered 2020-10-21, doi: 10.21468/SciPost.Report.2108

### Report

The manuscript “Position-Dependent Mass Quantum systems and ADM formalism”, by Davood Momeni

tries to present a Hamiltonian formulation of general relativity, inspired by position-dependent mass systems, and then it proposes a quantum version of the theory.

1) The Hamiltonian formulation of general relativity has been discussed extensively in the literature for decades. Some references are given, but many important contributions are missing (one can find them in textbooks or Review papers). In any case, the author should tone down his claims about his findings and their significance, and remove statements such as “this is the first time in literature when a first order Hamiltonian version of the gravitational field equations.”

2) The author’s approach is based on the “super mass tensor”. However, the author defines this quantity as a derivative on the Lagrangian, and hence expression (2.5) is a loop definition.

3) I cannot see how the complicated form of field equations (3.10) can be helpful.

4) The matter sector, which is crucial in GR since it is the source of non-trivial curvature and geometry, is missing from the discussion.

5) The author faces the problem as a simple quantum mechanical problem and not as a quantum field theoretical one, and thus the discussion on renormalizability etc is missing.

6) Solutions (5.7),(5.8) do not have an obvious meaning, and in any case it is strange that the author finds non-trivial structure in the absence of matter.

7) The English of the manuscript need editing.

In summary, a radical modification is needed before I will be able to reconsider the manuscript for publication.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -